Primulam

ABSTRACT

A board game incorporating the Ulam Spiral in which players compete to advance along a square spiral path of natural numbers via the prime numbers which are color coded according to the order of the subsequence of primes to which they belong. A player can occupy only one position on the board at any one time, and the player&#39;s position is determined by the prime index total that is accumulated via the roll of a die at each turn. The first player to reach the first prime number above some predetermined point on the natural number line wins the game.

FIELD OF THE INVENTION

This invention relates generally to board games, and more particularly to a board game where players compete to advance along a board's pathway according to chance and by rules established for prime number sequences of higher order.

BACKGROUND OF THE INVENTION

Board games exist with many different environments that include various modes and strategies for competing. Often in such games, the advancement of each individual player is dependent upon some sub-task involving chance, such as the rolling of a die or a drawing of a card to determine the range and/or direction of movement allowed along the game board's path toward the finish line.

BRIEF SUMMARY OF THE INVENTION

The disclosed invention relates to a game board consisting of the special arrangement of prime numbers discovered by the mathematician and scientist Stanislaw Ulam called the Ulam Spiral, or prime spiral. In the present board game, the set of all prime numbers arranged along the Ulam Spiral is subdivided into special subsequences of prime numbers which are distinguished one from another according to their order and corresponding color on the board. Advancement along the board path at each player's turn is determined by the prime number index that is rolled with the toss of a die. The number rolled is added to the index of a player's current prime number position on the board to determine the player's potential for advancement to the next prime number that can be occupied. The present board game may accommodate anywhere from two to six players. There are seven different colored sets of prime numbers on the playing board, all of which collectively represent the entire set of prime numbers within the range of natural numbers on the board path. There are certain rules of advancement that pertain to each colored subset of prime numbers.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.

FIG. 1—The PRIMULAM board game which includes the square spiral path of the natural numbers in the counterclockwise direction with the prime numbers on the board highlighted in the colors of Green, Red, Yellow, Blue, Silver, Purple, and Gold according to their order.

FIG. 2—An enlarged view of the center section of the PRIMULAM board game that includes only the first few inner spirals for the purpose of describing the mode of the invention via a short example provided herein.

FIG. 3—A list of the prime numbers up to 1009 and their corresponding indexes to use as a handy reference when playing the game.

FIG. 4—A list of the prime numbers up to 1009 which are categorized according to their color classification.

DETAILED DESCRIPTION OF THE INVENTION

Referring to FIGS. 1 and 2, the non-colored (or white-square) numbers on the game board are never occupied but exist only as the indexes of the Green prime numbers. For example, the number 1, or the first position in the center of the board, is the index for the first prime number 2 which is a Green occupiable position on the board. The next non-colored number is 4 which represents the index of the fourth prime number 7 which is also Green and therefore an occupiable position on the board.

The Green prime numbers can be occupied, and they also serve as the indexes of the Red prime numbers.

The Red prime numbers cannot be occupied but exist as the indexes of the Yellow prime numbers.

The Yellow prime numbers can be occupied, and they also serve as indexes of the Blue prime numbers.

The Blue prime numbers cannot be occupied but exist as the indexes for the Silver prime numbers.

The Silver prime numbers can be occupied, and they also serve as indexes of the Purple prime numbers.

The Purple prime numbers cannot be occupied but exist as the indexes of the Gold prime numbers which can be occupied.

Thus, the set of White and Red prime numbers combined form the indexes for the Green and Yellow prime numbers, the Green and Yellow prime numbers form the set of indexes for the Red and Blue prime numbers, and so on and so forth. An overview of the game mode and rules follow.

Movement in the counterclockwise direction along the spiral game board path is accomplished by advancing via the indexes of the prime numbers as displayed in the lower right-hand corner of each colored square. Board advancement at each turn is determined by the color of the prime number which would be landed upon when a die is rolled and added to the index of the player's current prime number index position on the board. The basic rules for advancement along the board path are:

Only the Green and Yellow prime number positions on the board can be occupied, and only one player can occupy one of those positions at any one time.

The Red, Blue, and Purple prime number positions cannot be occupied, and they have certain restrictions associated with them when their indexes are rolled.

The Blue prime numbers are a special subset of the Red prime numbers with an additional restriction placed upon a player's move when their index is rolled.

The Yellow prime numbers are a special subset of the Green prime numbers with a privilege associated with them if they are landed upon.

To begin, all players each roll a die to determine the order of play. If two or more people tie, then they re-roll to determine their order in turn.

When a player rolls a die, that number is added to the index of the player's current Green or Yellow prime number position on the board to reveal the index of the next higher prime number that is evaluated for occupation.

If on an initial turn, a player rolls a White number or a Red prime number, that represents the index of a Green or Yellow prime number board position (respectively) that can be occupied. Here are the options that apply to a player's initial roll as well as to future rolls which add to their current index total to produce an index of a Green or Yellow prime number:

1) If someone is already occupying the destination Green or Yellow prime number, then the player may opt to take possession of that Green or Yellow prime by bumping the current occupant up to the next higher unoccupied Green or Yellow prime. In such a move, the player bumps the current occupant of the destination Green or Yellow prime up and over any Red, Blue, or Purple prime number in the pathway to arrive at the next higher unoccupied Green or Yellow prime. If this move lands the bumped occupant on a Yellow prime, then the bumped occupant may advance to the next higher unoccupied Green prime number, or they may remain on the Yellow prime. For example, a player may choose to remain on a Yellow prime rather than advance to the next higher unoccupied Green prime in the hope that a subsequent roll of a “1” or a “6” of the die would provide that player with an advantageous diagonal move which will be explained later. Finally, if the occupant is being bumped off of a Yellow prime, then the player doing the bumping may advance to the next higher Green or Yellow prime number after the bumped occupant is relocated.

2) Or, a player may occupy the next lower unoccupied Green or Yellow prime available if that player chooses not to bump an occupant out of a Green or Yellow destination prime.

3) Or, if the new index total lands a player on a Yellow prime, then that player can jump ahead to the next higher unoccupied Green prime or choose to remain on the Yellow prime. The jump (if chosen), includes the privilege of passing over all Red, Blue, Purple, and any occupied Green or Yellow prime numbers in the pathway to arrive at the next higher unoccupied Green or Yellow prime.

4) Or, the player can forfeit the roll (for example, in anticipation of a future diagonal move opportunity).

If on a player's initial turn, he or she rolls a Green prime number, that prime number represents the index of a Red prime number position which cannot be occupied. The player then has the following options that apply to their initial roll as well as to future rolls which add to their index total to produce an index of a Red prime number:

1) The player can move to any unoccupied Green or Yellow prime number below the Red prime number. If the player is already occupying a Green or Yellow prime number immediately below the Red prime number, then that player is stuck for that round.

2) If there is an occupied Green or Yellow prime number below the Red prime number index that the player rolled (such as in a first roll for example), the player can bump the occupant of that lower Green or Yellow prime number up and over any Red, Blue, or Purple primes as well as over any occupied Green or Yellow primes to enable the bumped player to occupy the next higher Green or Yellow prime number. If the player that was bumped up lands on a Yellow prime, then the rules of landing on a Yellow prime apply to that player; i.e., that player can opt to either move up to the next higher unoccupied Green or Yellow prime number by jumping over all occupied Green and Yellow primes as well as any Red, Blue, or Purple primes in the pathway, or that player can remain on that Yellow prime in anticipation of a future diagonal move opportunity.

3) Or, the player can forfeit the roll (for example, in anticipation of a future diagonal move opportunity).

The only times that a player can pass over a Red prime number are:

1) When a number is rolled to add to the index of the player's current prime number position to produce a new index that exceeds that of the next Red prime number in the path (i.e., “rolling”).

2) Or, when the occupant of a Green or Yellow prime number immediately below the Red prime number is bumped up and over a Red prime by another player to occupy a higher Green or Yellow prime number position (i.e., “bumping”).

3) Or, when a player lands on one of the Yellow primes on the board and that player opts to jump all consecutively occupied Green and Yellow primes, as well as any Red, Blue, or Purple prime numbers that may be in the pathway, to occupy the next higher unoccupied Green or Yellow prime (i.e., “jumping”).

If a player rolls an index total equal to a Yellow prime number, then that is the index of a Blue prime number on the board. If on a player's first roll, he or she rolls the index of the first Blue prime number (i.e., the index of the prime number 11 which is 5), then the player is constrained from moving off of the starting block for that round. But if during the game, a player's index total sums up to 59, that is the index for the other Blue prime number 277 on the board and the player is bumped backward to the second lower unoccupied Green Prime number.

If a player rolls an index total equal to the Silver prime number, 31, then that is the index of the Purple prime number 127 on the board. If during the game a player's index total sums to 31, then the player is bumped backward to the third lower unoccupied Green Prime number.

If a player rolls an index total equal to 11, this is the index for the Silver prime number 31. Landing on the Silver prime number entitles the player to move forward to the second higher unoccupied Green or Yellow prime number.

If a player rolls an index total equal to 127, this is the index for the Gold prime number 709. Landing on the Gold prime number entitles the player to move forward to the third higher unoccupied Green or Yellow prime number.

If the player lands on a Green or Yellow prime number and there is higher Green or Yellow prime number connected diagonally to that Green or Yellow prime, then on the next roll the player can advance to the higher diagonal Green or Yellow prime number if the player rolls a 1 or a 6. Bumping another occupant upward out of the destination diagonal Green or Yellow prime according to the rules applies here. Also, the rule for landing upon a Yellow prime applies to a diagonal move if the diagonal destination prime is a Yellow Prime number (i.e., the player can move forward to the next unoccupied Green prime number after making the move to the diagonal Yellow prime number).

If a player lands on a Green or Yellow prime and there is higher Red prime number connected diagonally to the Green or Yellow prime number landed upon, then on the player's next roll he or she can advance to the next lower unoccupied Green or Yellow prime number below that Red prime number if the player rolls a 1 or a 6. Bumping another player out of a lower Green or Yellow prime number below the diagonal Red prime number is allowed in this scenario.

Anytime a bump forward encounters other occupied positions, then the person being bumped can jump over all occupied Green and Yellow primes as well as any Red, Blue, or Purple primes in the pathway to occupy the next higher unoccupied Green or Yellow prime. If the bumped player lands on a Yellow prime, then the rules for landing on a Yellow prime apply.

On the final stretch to the ending prime 1009, the final index total that that player rolls has to land that player exactly on the index of 169 in order for that player to win the game. An index total any more or less than 169 will require the player to pass until their rolled index total equals 169.

The 139 Green, Yellow, Silver, and Gold primes on the board which include the prime numbers (2, 5, 7, 13, 19, 23, 29, 31, 37, 43, . . . ) represent the OEIS prime number sequence A333242 discovered by the inventor of this game which represents the P′ subsequence of the set of all prime numbers P.

The nine Yellow, Silver, and Gold primes on the board which include the prime numbers (5, 31, 59, 179, 331, 431, 599, 709, 919) represent the prime numbers that have a Red, Blue, or a Purple prime number as their index. These prime numbers correspond to the OEIS prime number sequence A333243 discovered by the inventor of this game which represent the P′″ subsequence of the set of all prime numbers P. If a player lands on one of these primes, then that player can perform a “jump” to occupy a higher unoccupied Green or Yellow prime number position on the board according to the rules associated with each color.

The three Blue and Purple primes (11, 127, 277) represent the prime numbers with a Yellow or a Silver prime number as their index. These prime numbers correspond to the OEIS prime number sequence A333244 discovered by the inventor of this game which represent the P″″ subsequence of the set of all prime numbers P. If a player lands on one of these primes, then that player is impeded in their advancement for that round according to the rules associated with each of those colors.

To summarize the game board moves that are allowed versus those not allowed, if the player's index total lands the player on the following colors (listed below in order of their frequency of occurrence on the board), then:

Green—the position can be occupied.

Red—the player can't land on this prime but can occupy the next lower unoccupied Green or Yellow prime number position or can land on the Silver prime number position and reap the special privilege associated with that color prime. The rules for bumping may apply here depending upon the situation.

Yellow—the player can move forward to the next higher unoccupied Green or Yellow prime number position by jumping anyone in the pathway, or the player can remain in this position.

Blue—if on the first roll of the die (i.e., a prime index of 5 for the prime number 11), then the player can't move and is stuck for that round; if for the other Blue prime on the board (i.e., an index total of 59 for the prime number 277), then that player is displaced to the second lower Green or Yellow prime number below that Blue prime.

Silver—the player can move forward to the second higher unoccupied Green or Yellow prime number by jumping anyone in the pathway.

Purple—the player is bumped back down from their currently occupied position to the third lower unoccupied Green or Yellow prime number.

Gold—the player can move forward to the third higher unoccupied Green or Yellow prime number position by jumping anyone in the pathway.

As a note, the combined set of the Green, Yellow, Silver, and Gold prime numbers advance at a rate slightly greater than the frequency of the set of all prime numbers; and the combined set of the Red, Blue, and Purple prime numbers advance at a rate slightly greater than the set of twin primes.

It is noted that the present board game is deemed to have educational value as a tool to aid the student in the learning and memorization of the first 169 prime numbers and their corresponding indexes. Thus, variations of the game may include omitting the prime number indexes from the box of each prime number, or omitting the prime numbers themselves and leaving just the indexes, as a challenge to the student to learn the index of each prime number or to know which prime number is associated with each index.

Mode of Operation of the Invention

Referring to FIGS. 1 and 2 of the board game, each colored square includes a prime number in the upper left-hand corner of the square with that prime number's index in the lower right-hand corner of the square. A player's counterclockwise advancement along the square spiral path of natural numbers is determined by the roll of a die which adds to the player's current index number position on the board. Thus, the physical position of a player's game piece on the board is determined by his or her index total, and the player's board or “score” position is determined by the prime number which is being occupied. As an aid to navigate the game board, FIG. 3 is a chart of all the prime numbers up to 1009 with their corresponding index (or sequence) within the set of all prime numbers. As a reference, FIG. 4 is a list of all the prime numbers on the game board which have been categorized according to their color classification. A review of the attributes of each prime number color follows:

The numbers in the White squares are never occupied but exist only as the indexes of the Green prime numbers.

The Green prime numbers can be occupied, and they also serve as the indexes of the Red prime numbers.

The Red prime numbers cannot be occupied but exist as the indexes of the Yellow prime numbers.

The Yellow prime numbers can be occupied, and they also serve as indexes of the Blue prime numbers.

The Blue prime numbers cannot be occupied but exist as the indexes for the Silver prime numbers.

The Silver prime numbers can be occupied, and they also serve as indexes of the Purple prime numbers.

The Purple prime numbers cannot be occupied but exist as the indexes of the Gold prime numbers which can be occupied.

It is expedient to simulate the beginning of a game to introduce a few of the basic rules and how they are applied. In this simulation, we will assume two players. Referring to FIG. 2, let's say Player #1 rolls the number 1 with his or her die. That player then advances to the first prime number 2 which is indicated in the upper left-hand corner of the first Green square to the right of the starting block because the index number 1 was rolled, and 1 is the index for the prime number 2. Note that a move is determined by the prime number index that is rolled with the die. Player #2 then rolls his or her die and also comes up with a 1. Since the prime number 2 position is already occupied, Player #2 opts to bump Player #1 forward out of prime number 2 so that Player #2 can occupy the prime number 2. That enables Player #1 to move forward to the next higher Green or Yellow prime number on the board which in this case would be either the prime number 5 (Yellow) or the prime number 7 (Green). Since Player #1 chooses the higher prime number 7, that player's index total is now the index of the prime number 7 which is 4, meaning the 4^(th) prime number. Note that had Player #1 desired, he or she could have opted to land on the Yellow prime number 5 in the hope that an advantageous diagonal move might be possible with a future roll of either a 1 or a 6 of the die. For example, if Player #1 had opted to land on the Yellow prime number 5, and if Player #1 subsequently rolled a 1 or a 6 on the next roll of the die, Player #1 would have the opportunity to move forward diagonally to occupy the prime number position of 19 which would have enabled him or her to skip over prime numbers 7, 11, 13, and 17 along the spiral path. Next, say that Player #2 rolls another 1 with the die. Player #2 then adds that roll of 1 to his or her current index total (the index of the prime on which he or she is currently residing) to get a new index total of 2. But since the index 2 is the index for the prime number 3 which is a Red prime, the rules state that since that Red prime can't be landed upon, then the player has the opportunity to occupy the next lower unoccupied Green or Yellow prime on the board. But since Player #2 is already on the only Green or Yellow prime number on the board less than the prime number 3, that player is stuck there for that round. However, note that had there been any other unoccupied Green or Yellow prime numbers on the board greater than the prime number that Player #2 was sitting on but less than the destination Red prime, then Player #2 would have been able to advance to that higher Green or Yellow prime number. Or, if there had been an occupied Green or Yellow prime number less than the Red prime index rolled, then Player #2 would be allowed to bump another player out of that Green or Yellow prime number to occupy that position. In that case, the player being bumped over the Red prime number will be able to do so to occupy the next unoccupied Green or Yellow prime number position according to the rules previously established.

Now consider a hypothetical point later in the game where Player #1 resides on the Green prime number 23 (index position 9). Say that Player #1 rolls a 1 with the die. That gives Player #1 a new index total of 10. Now assume that Player #2 is occupying the Green prime number 29 which has the index of 10. Player #1 can opt to “bump” Player #2 out of the Green prime number position 29 to occupy that prime number, but the consequence will be that Player #2 will be able to jump forward to occupy the next higher available prime number which in this case is 43. This is made possible by Player #2 being bumped forward to the Silver prime number 31 and was therefore entitled to move forward two Green or Yellow positions which included jumping over all colored primes in the pathway to occupy that second higher position. So, by being bumped by Player #1, Player #2 just advanced 14 spaces on the board. And in this case, Player #2 was able to jump over the Red prime number 41 to do so. As another option, Player #1 could have taken that roll of 1 and advanced diagonally to the higher Green prime number 47 (with an index of 15) which would have been a highly more advantageous move.

The short scenario above included the actions of bumping, jumping, and being stuck in a prime number position in a round. Many other various situations will result, especially as the number of players increase. In all cases, following the rules as previously set forth will determine the options and allowable actions by each player. It has been observed that there is no noticeable advantage to a player starting in the game either first or last or somewhere in between, as the element of chance via the rolling of the die combined with the rules associated with a player's advancement or impediment in progressing along the game path equalize the player's chances of winning the game regardless of starting position. 

What is claimed is:
 1. A board game comprising: a game board that incorporates the Ulam Spiral as the pathway of natural numbers along which a player advances via the prime numbers on the path.
 2. A board game comprising: the board game in claim 1 wherein the prime numbers and/or their associated indexes are enclosed within squares or other shapes that are filled or otherwise marked with a color according to the prime number subsequence to which they belong.
 3. A board game comprising: the board game in claim 2 wherein the prime number subsequences identified by color code on the game board include the inventor's prime number sequences A333242, A333243, and A333244 which are published by the Online Encyclopedia of Integer Sequences.
 4. A board game comprising: the board game in claim 1 which includes a game board positional token for each player that is color coded to identify each individual player and their location on the board.
 5. A board game comprising: the board game in claim 4 which includes a colored die for each player that matches the color of the player's positional token to keep track of which player has rolled a die and which token gets moved.
 6. A board game comprising: the board game in claim 1 which includes a reference chart for each player of all the prime numbers up to 1009 and their associated indexes or sequence within the set of all prime numbers.
 7. A board game comprising: the board game in claim 1 which includes a quick guide of the rules associated with the various allowable moves and restrictions according to the prime number which would be landed upon as determined by the roll of a die.
 8. The name PRIMULAM of the board game in claim 1, wherein: in one sense, a compound word consisting of “prim-” representing “prime numbers”, and “-ulam” representing the last name of the discoverer of the Ulam Spiral, the mathematician and scientist Stanislaw “Ulam”; in another sense, “primulam” is the accusative feminine singular of “primulas”, meaning “the very first” in Latin.
 9. A board came comprising: the board game in claim 2 as an educational tool to aid the student in the learning and memorization of the first 169 prime numbers and their associated indexes. Variations of the game board may include omitting either the prime numbers or the indexes of the prime numbers from the colored squares as a challenge to the student to memorize the first 169 prime numbers and their associated indexes.
 10. A board came comprising: the board game in claim 1 which may be modified to configure different strategies, challenges, privileges, or restrictions in order to enhance the game.
 11. A board came comprising: the board game in claim 1 which may include any computerized variations of the game which would increase the entertainment and/or ease of playing the game such as lighted colored squares, sound effects, push button controls, or electronic methods of simulating the process of a random roll of the die and/or the identification of the location of a player on the board. 